The use of specially shaped optical pulses for the generation of certain types of THz electromagnetic waveforms has been explored to a limited extent. In Ref. [1], pairs of identical, though phase-controlled, subpicosecond optical pulses were used to initiate and then coherently terminate charge oscillations in a biased double quantum well (QW) to generate THz waveforms consisting of a predetermined number of THz cycles. This technique is limited to creating narrowband THz radiation whose frequency is given by the energy splitting of the states responsible for the charge oscillations; the maximum duration of the THz emission generated in this fashion is limited by the dephasing time between these states, which also determined the bandwidth of the THz radiation. In Refs. [2,3], the use of trains of subpicosecond optical pulses was examined to generate trains of THz pulses via optical rectification in a biased QW. In that work, the individual THz pulses generated were essentially single cycle. Moreover, the length, repetition rate, and strength of such pulse trains is constrained by the saturation of the QW with carriers. Another demonstrated technique involves interfering chirped optical pulses on a photoconductive source;[4] linear chirp results in narrowband THz emission. In these sysytems, saturation is determined by space-charge effects as well as perhaps screening of the bias by the emitted THz field. Other applications of optical pulse shaping for the generation of programmable THz waveforms are disucssed in Ref. [5].

In the following, we discuss a technique based on optical rectification in a dc-biased QW for the generation of quasi-half-cycle THz pulses using specially tailored optical pulses.[6–8] One class of optical pulses considered contains zero area, which leads to the minimum possible net transfer of energy from the pulse to the medium.[9,10] Saturation of the QW with carriers by the optical pulse can thereby be largely circumvented,[11–14] thus leading to the potential for the generation of THz pulse trains whose overall duration, strength, and repetition rate can exceed those using the technique of Refs. [2,3].

Our immediate motivation in exploring the generation of half-cycle THz pulses is to probe coherent electron-hole wavepacket dynamics in semiconductor heterostructures,[15] and to pursue the high-field physics of these wavepackets in their response to impulsive electromagnetic fields along the lines of recent work in atomic physics.[16] In addition, trains of half-cycle THz pulses will be useful for coherent control problems, such as guiding electron-hole wavepacket evolution towards desired final states and for molecular problems.

We begin by reviewing the physics which determines the optical rectification signal from a single biased QW using shaped optical pulses. We then explore the ramifications of the resulting model for the generation of quasi-half-cycle THz pulses. We follow a two-step procedure to model the coherent optical and THz dynamics. [1,16] To treat the exciton population, we employ the coupled Maxwell and semiconductor Bloch equations (SBE) in the low-density limit projecting out only the lowest-lying heavy-hole exciton. The generalization to several resonances is straightforward; however, in order not to obfuscate the basic physical issues, we consider here a single resonance. For this to be valid, we assume that the optical pulse spectrum primarily overlaps the optical transition from the crystal ground state to the 1s heavy-hole exciton formed from the subbands nearest the bandedge. We first obtain in linear optics the dynamical dipolar response at optical frequencies. We then obtain the associated interband polarization P _opt(t) induced by the optical field, which determines the coherent part of the exciton population through the SBE. Second, we calculate the time-dependent dipole moment P _THz(t) at THz frequencies,[1] which in turn leads to the THz electric field via 𝜀 _THz(t)∝P̈ _THz(t). Throughout we assume a single dc-biased QW of thickness L_z ≪ƛ=ħc/(E _ex n_b ) (e.g., ∼400 Å for a GaAs/AlGaAs QW) where ƛ is the reduced wavelength in the medium of light associated with the exciton resonance, E _ex is the exciton energy measured with respect to the crystal ground state, c is the speed of light in vacuuo, and n_b (∼√11 in GaAs) is the background refractive index.

From the theory of the nonlocal optical response of a QW to monochromatic light at normal incidence (optical field at normal incidence, i.e., excitation wavevector in the QW plane k _∥ =0), the interband polarization density P _opt(ε,z) induced by an incident electric field 𝜀_inc(ε, z′) oscillating at optical frequency ε/ħ is[17]

where g is a spin-orbit factor (=1 for heavy-hole excitons), eμ is the dipole matrix element between the s-like conduction- and p-like bulk valence-band Bloch states, F _ex(r _∥) is the quasi-two-dimensional exciton envelope function, f_c (f_υ ) is the envelope function for the conduction (valence) subband of interest, and Γ_sc (typical values given below) is a phenomenological nonradiative damping. The dephasing time $T_2^{*}$ is related to Γ_sc by Γ_sc/ħ=1/$T_2^{*}$. Γ_sc accounts for scattering, which either directly destroys the phase of the excitation (pure dephasing) or scatters the state to a wavevector k _∥≠0, or inhomogeneous broadening. These excitons contribute to the incoherent part of the population (see below) and thus do not further interact coherently with the incident light field. In this study we neglect excitation-induced dephasing. 𝜀 _tot(ε,z′) is the dependence in the direction normal to the QW plane of the total electric field-both incident and induced. The areal interband polarization density is P _opt(ε)= ${\int }_{-\infty }^{\infty }$ dz P _opt(ε, z). Since by assumption L_z ≪ƛ, 𝜀_tot(ε,z) is essentially constant across the QW. From Eq. (1), we therefore obtain

where z _QW is the position of the QW and S = ${\int }_{-\infty }^{\infty }$ ${\mathit{\text{dz}}f}_c^{*}$(z)f_υ (z) is the overlap integral of the conduction- and valence-subband envelope functions. This allows us to relate the incident field to the total field as 𝜀_tot(ε, z _Qw)= t(ε)𝜀_inc(z _Qw) with t(ε) the transmission coefficient for the electric field through the QW: [19–21] t(ε)=(ε-E _ex+iΓ_sc)/(ε-E _ex + iΓ). Here Γ = Γ_sc+Γ_rad and Γ_rad is the radiative width of k _∥=0 excitons obtained from the theory of QW exciton polaritons.[22] Combining the previous results gives

with χ(ε) = ge ² μ ²|F _ex(0)|²|S|²/(ε - E _ex + iΓ) the susceptibility relating P _opt(ε) and 𝜀_inc(z _QW).

Since we are interested in the response of the system not to a monochromatic field but to a pulse, we rewrite Eq. (3) in the time domain:

with 𝜀_inc(t) = A(t) exp(-iω ₀ t) the incident electric field at the location z _QW of the QW, A(t) the pulse envelope, and ω ₀ the carrier frequency. The susceptibility χ(t) = -ige ² μ ²|F _ex(0)|²|S|² exp[-i(E _ex - iΓ)t/ħ]θ(t) with θ(t) the Heaviside step function. Hence

with β ₁ = -ige ² μ ²|F _ex(0)|²|S|². Note that apart from the dephasing factor appearing in the integral, a pulse of vanishing area [t→∞ in Eq. (6)] coherently depopulates the QW following its passage,[11,13,14,23] i.e., 𝛲(∞)=0.

To proceed, we need to establish the relationship between the polarization and the density. We write N(t) = N _coh(t) + N _incoh(t) where N _coh(t) and N _incoh(t) are the coherent and incoherent parts of the population per unit area, respectively. Only the coherent part interferes with the optical field although the total population N(t) modulates the dipole at THz frequencies. From the SBE, we have [24]

Thus N _coh(t) is governed by the optical pulse; it is here we can attain control of the population through the pulse envelope A(t).

We can now relate the incident pulse with the THz radiation ultimately produced. For the case in which the laser excitation spot is much smaller than the wavelengths of the relevant THz frequencies, the dipole moment giving rise to the THz-frequency radiation is P _THz(t) = dN(t) with d = e${\int }_{-\infty }^{\infty }$ dz z[|f_c (z)|² - |f_υ (z)|²], and the THz electric field is 𝜀_THz(t)= -(c ² r)^-1 P̈_THz(t). The time dependence, note, of 𝜀_THz(t) is independent of the dipole approximation (applied to the THz field). We assume that N _coh is not significantly depleted by $T_2^{*}$ during the pulse. This allows us to dispense in our calculation of 𝜀_THz(t) with dephasing while the pulse is incident on the QW; we assume that the duration of A(t) is much less that ħ/Γ so that the time derivatives of P _THz(t) will be dominated by N _coh(t). This gives

In the following we shall consider the THz radiation resulting from three pulse forms. The first two will be pulses of nonzero area. We shall consider a symmetric pulse given by a Gaussian envelope A(t)= e ^-t2/τ2 (type I), and an asymmetric pulse A(t) = [1- tanh(t/σ]e ^-t2/τ2 with σ≪τ (type II). For the type-II pulse, τ controls the temporal rise and σ the fall. The third pulse shape we consider is A(t)= -tanh(t/σ)e ^-t2/τ2, again with σ≪τ (type III). In this case, the pulse has zero area and is odd in t; the role of the hyperbolic tangent is to produce a rapid π phase change in the pulse center over a time σ. Note, that by using phase-mask technology and sufficiently short seed pulses, σ can be in the tens of fs range, while τ might be in the hundreds of fs to ps range. (Recall, we want τ to be as small as possible compared with ħ/Γ; however, we want to avoid too much spectral overlap with more than the lowest-lying exciton level.)

 

Fig. 1(a). A(t) for a type-I pulse.

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Fig. 1(b). 𝛲²(t) for a type-I pulse.

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Fig. 1(c). ℱ(t) for a type-I pulse.

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Figure 1 shows (a) A(t), (b) 𝛲²(t) [which is proportional to N _coh(t)], and (c) ℱ(t) for a type-I pulse. This figure illustrates the difficulty associated with trying to use such simple optical pulse forms to generate quasi-half-cycle THz pulses. Note that the comparable curvature in the levelling off of N(t) compared with its rise leads to a single-cycle THz pulse in which the two lobes of opposite polarity are of the same amplitude. This suggests considering highly asymmetric pulse shapes. Apart from this, however, as Fig. 1(b) shows, this pulse shape leaves a population of excitons in the QW after its passage. In Fig. 2 are plotted (a) A(t), (b) 𝛲²(t), and (c) ℱ(t) for a pulse of type II. We have chosen σ = 0.1τ. Here, the amplitude of the lobe of the THz field associated with the falloff of A(t) is roughly ten times that of the lobe associated with its rise. [Note that the area under 𝜀_THz(t) must vanish since for reasonable pulse shapes A(t), 𝛲̇(t) goes to zero as t→±∞.] As can be seen in Fig. 2(b), however, this pulse also leaves population in the QW following the passage of the pulse.

 

Fig. 2(a). A(t) for a type-II pulse. σ =0.1τ.

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Fig. 2(b). 𝛲²(t) for a type-II pulse. σ =0.1τ.

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Fig. 2(c). ℱ(t) for a type-II pulse. σ =0.1τ.

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A solution to both these problems (the generation of a net exciton population and the failure of symmetric pulses to give rise to strong electromagnetic lobes of one sign without producing similar lobes of the other) is obtained using a type-III pulse which is of vanishing area, as shown in Fig. 3(a). This pulse is designed to depopulate coherently the QW upon its passage, according to Eq. (6). This is indeed seen to occur in Fig. 3(b). Moreover, the rapid sign change of the envelope near t=0 produces a THz signal [Fig. 3(c)] with a strong high-frequency lobe of one sign and flanked by weak low-frequency lobes of opposite sign.

 

Fig. 3(a). A(t) for a type-Ill pulse. σ =0.1τ.

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Fig. 3(b). 𝛲²(t) for a type-III pulse. σ =0.1τ.

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Fig. 3(c). ℱ(t) for a type-III pulse. σ =0.1τ.

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As mentioned in the introduction, our interest in type-III optical pulses is to generate trains of intense quasi-half-cycle THz pulses at high repetition rates. A key issue is to avoid saturating the QW with excitons by the optical pulses; type-III pulses are optimal in achieving this aim. It is important therefore to ascertain quantitatively to what degree the use of type-III pulses confers this advantage over pulses of type II by comparing the background population of excitons remaining in the QW after the passage of each type of pulse. For purposes of comparison, we shall assume that σ and τ are the same for both cases. Also recall σ≪τ. Under these assumptions, quasi-half-cycle THz pulse of similar durations are produced in both cases and the peak electric field of 𝜀_THz associated with both pulses are comparable.

The foregoing theoretical treatment provides a detailed basis for the comparison; however, simple estimates amply demonstrate our main point. For the type-II pulse, the population of excitons is generated via Eq. (7); thus for the pulse of type II, we are interested in N _coh(∞). For the type-III pulse, N _coh(∞) vanishes by design; instead, the relevant population is generated via $T_2^{*}$ over the duration τ of the pulse which feeds N _incoh. Simple estimates give that the ratio of the populations remaining in the QW following the passage of a type-III versus a type-II pulse is τΓ_sc which by assumption is much less than unity. For example, if τ = 500 fs and ${\mathrm{\Gamma }}_{\text{sc}}^{-1}$ = 5 ps, then this ratio is 0.1. Consequently, saturation of the QW under high-repetition-rate operation may be largely circumvented by using optical pulses of type III, since the background population of carriers can be reduced by a factor of τΓ_sc. Note, this discussion leaves out the possibility of device saturation via the emitted THz field-a problem discussed more thoroughly in Ref. [5] and the references cited therein.

Before concluding, we would like to comment on some practical limitations on the values of τ and σ for the scheme discussed here to work in real materials. First, we need τ, σ≪$T_2^{*}$. In III-V QW’s at low temperature and density, $T_2^{*}$ is in the 7 ps range; however, $T_2^{*}$ increases with both temperature and density. The other constraint on τ and σ concerns the need to avoid insofar as possible saturating the QW via excitation of higher-lying levels (e.g., 2s, 3s, etc. excitons and continuum electron-hole pairs). For narrow QW’s, say of thickness not exceeding 100 Å in III-V’s, the 1s–2s splitting is in the 10 meV range (taking the exciton binding energy as a rough figure). Thus, the fastest timescale present in the optical pulse envelope, i.e., σ, should not be less than ∼100 fs. If for example we take σ = 100 fs, then for τ = 1 ps we still have τ, σ≪$T_2^{*}$, but not by a great deal. Two ways of circumventing this problem are as follows. One way is simply to shape the optical pulse in such a way as to suppress the spectral overlap with the higher-lying transitions. This will produce a pulse pedestal, but the effect on the THz waveform will be small it depends on the square of the optical excitation field. Of course, it will be necessary to optimize the pulse shape in order to minimize the net population generated subject to constraints associated with the desired THz pulse shape. The second possibility is to use a wider-gap material, such as a II-VI QW in which the exciton binding energy is in the 40 meV range. Such materials are attractive since they may also facilitate the generation of higher frequency THz signals.

To conclude, we have investigated the generation of quasi-half-cycle THz pulses utilizing specially shaped optical pulses via optical rectification in a biased QW. We have shown that high-repetition-rate trains of such pulses can be facilitated by using optical pulses designed to depopulate the QW coherently of excitons following the passage of the pulse. We have neglected optical nonlinearity effects such as phase-space filling and excitation-induced dephasing. These may be incorporated by using the full Maxwell-semiconductor Bloch equations or phenomenologically, by assuming density dependent quantities β ₁ (oscillator strength) and Γ_sc (excitation-induced dephasing). Provided the optical pulses are not too short, it should suffice to assume temporally local density dependent quantities.

We are grateful to T. B. Norris and S. Tomsovic for helpful conversations. This work was supported by the Office of Naval Research.